O nelinearnih valovih : delo diplomskega seminarja

Abstract
V delu diplomskega seminarja se posvetimo posameznim nelinearnim parcialnim diferencialnim enačbam različnih redov. Najprej predstavimo metodo reševanja kvazilinearnih parcialnih diferencialnih enačb, metodo karakteristik in izrek, ki pod določenimi pogoji zagotavlja lokalno enolično rešljivost teh enačb. Na primeru Eulerjeve enačbe predstavimo problem neenoličnosti globalne rešitve pri reševanju z metodo karakteristik. Enačbo zapišemo v obliki ohranitvenega zakona in predstavimo rešitev s pomočjo udarnih valov. V nadaljevanju se posvetimo enačbam z disperzijo in vplivu disperzije na obliko udarnih valov. Predstavimo rešitev Burgersove enačbe s pomočjo Hopf-Coleove transformacije. Delo zaključimo s predstavitvijo preproste, enosolitonske rešitve Korteweg-deVriesove enačbe.

Language: Slovenian metoda karakteristik, transportna enačba, ohranitveni zakon, udarni val, Burgersova enačba, Korteweg-deVriesova enačba, soliton Final seminar paper 2.11 - Undergraduate Thesis FMF - Faculty of Mathematics and Physics 2018 20.500.12556/RUL-103652 517.9 18457177 21.09.2018 1030 327 Copy citation

## Secondary language

Language: English On nonlinear waves This thesis is dedicated to the analysis of certain nonlinear partial differential equations of different orders. We begin with the method of characteristics. We show how the method is used for solving quasilinear partial differential equations. We also state the existence theorem which tells us under which conditions there exists a unique local solution. Using the example of Euler equation, the problem of non-uniqueness of the solution, constructed by means of the characteristics, is presented. The equation is written in the form of a conservation law and the solution is constructed by introducing the shock waves. We continue by focusing on the equations with dispersion and we study its impact on the shape of shock waves. We present the solution of Burgers equation using the Hopf-Cole transformation. The thesis is concluded by the presentation of a simple singlesoliton solution of the Korteweg-deVries equation. method of characteristics, transport equation, conservation law, shock wave, Burgers' equation, Korteweg-deVries equation, soliton

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